A humming sound from an air conditioner that won't turn on is usually a sign of an electrical problem. There are many electrical components that can contribute to this issue, but the primary suspects are your fuses, capacitors, and fan motor.
Replace the Fuses
Your air conditioner's outdoor unit is protected by fuses that take the brunt of electrical damage in the event of something like a circuit overload. Fuses are relatively inexpensive components, which makes them cost-effective tools to protect much more pricey components.
If your fuses are faulty, whether by old age or electrical malfunction, your air conditioner may still be receiving some power, but not enough to start and run normally. This will cause it to hum or buzz, as the air conditioner is technically receiving power, but remain inoperable.
Fuses are simple parts to replace, though caution is advised. To be safe, shut off the circuit that powers your air conditioner before doing any work. You can use a multimeter to test your current fuses or replace the current fuses if you have spares. If you don't feel comfortable trying this yourself, call a professional for repairs.
Have the Capacitors Replaced
Capacitors are responsible for starting your air conditioner and keeping it running. In order to get the fan blades moving, your air conditioner essentially requires an extra push or boost or power. If your air conditioner lacks this push, it may have the ability to run, but not enough to actually start it.
One way to test this is by taking a stick and manually pushing the fan blades into motion. If they gradually speed up and run properly, the issue is most likely the start capacitor no longer working correctly. This will need to be replaced, so contact a professional for service. If moving the fan manually doesn't help, the issue could still be with the run capacitor instead of the start capacitor but will be harder to diagnose without a technician.
Diagnose the Fan Motor
In more serious cases, the fan motor itself could be faulty. If the fan motor is faulty, your air conditioner will not work even if everything else is fine. You may hear sounds like rattling and banging coming from the outdoor unit or the fan blades may be turning very slowly, but the absence of these symptoms doesn't necessarily mean the motor is fine.
If the other solutions didn't work, the motor is the next most likely culprit. While this is a more complex component, it will rarely require replacing your entire air conditioning system. An air conditioning repair technician will be able to replace the motor for you and restore your air conditioner to good working order.